Hokkaido Mathematical Journal ol. 45 (2016) p. 271 291 Finiteness of the Moderate Rational Points of Once-punctured Elliptic Curves uichiro Hoshi (Received February 28, 2014; Revised June 12, 2014) Abstract. In the present paper, we prove the finiteness of the set of moderate rational points of a once-punctured elliptic curve over a number field. This finiteness may be regarded as an analogue for a once-punctured elliptic curve of the well-known finiteness of the set of torsion rational points of an abelian variety over a number field. In order to obtain the finiteness, we discuss the center of the image of the pro-l outer Galois action associated to a hyperbolic curve. In particular, we give, under the assumption that l is odd, a necessary and sufficient condition for a certain hyperbolic curve over a generalized sub-l-adic field to have trivial center. Key words: moderate point, once-punctured elliptic curve, hyperbolic curve, Galoislike automorphism. Introduction In the present paper, we discuss the finiteness of the set of moderate rational points of a hyperbolic curve over a number field. First, let us review the notion of a moderate point of a hyperbolic curve. Let l be a prime number, k a field of characteristic zero, k an algebraic closure of k, and a hyperbolic curve over k. Write G k = Gal(k/k), for the prol geometric fundamental group of [i.e., the maximal pro-l quotient of π 1 ( k k)], and ρ : G k Out( ) for the pro-l outer Galois action associated to. In [6], M. Matsumoto studied a k-rational point x (k) of that satisfies the following condition E(, x, l) [cf. [6, Introduction]]: E(, x, l): If we write s x : G k π 1 () for the outer homomorphism induced by the k-rational point x (k), then the kernel of the composite 2000 Mathematics Subject Classification : 14H30.
272. Hoshi G k s x π1 () Aut( ) where the second arrow is the action obtained by conjugation coincides with the kernel Ker(ρ ) of the outer action ρ : G k Out( ). For instance, Matsumoto proved that, roughly speaking, there are many hyperbolic curves over number fields which have no rational point that satisfies the above condition E(, x, l) [cf. [6, Theorem 1]]. As in [5], in the case where k is a number field, we shall say that a k- rational point x (k) of is l-moderate if the above condition E(, x, l) is satisfied [cf. [5, Definition 2.4, (i)]; the equivalence (1) (3) of [5, Proposition 2.5]]; moreover, we shall say that a k-rational point of is moderate if it is p-moderate for some prime number p [cf. Definition 2.1]. Typical examples of moderate points of hyperbolic curves are as follows: (a) A closed point of a split tripod [i.e., P 1 k \ {0, 1, } ] corresponding to a tripod p-unit [cf. [5, Definition 1.6]] [that is a certain higher circular p- unit cf. [5, Remark 1.6.1]] for some prime number p [cf. [5, Proposition 2.8]]. (b) A torsion point of [the underlying elliptic curve of] a once-punctured elliptic curve whose order is a [positive] power of a prime number [cf. [5, Proposition 2.7]]. (c) Every Q un-l -rational [cf. [5, Definition 1.8]] point of the [compactified] Fermat curve over Q of degree l [i.e., Proj(Q[s, t, u]/(s l + t l + u l )) ] if l is 5 and regular [cf. [5, Remark 3.5.1]]. Let us recall from [5, Remark 2.6.1, (i)], that the notion of a moderate point of a hyperbolic curve may be regarded as an analogue of the notion of a torsion point of an abelian variety. On the other hand, it is well-known that the set of torsion rational points of an abelian variety over a number field is finite. Thus, we have the following natural question: Is the set of moderate rational points of a hyperbolic curve over a number field finite?
Finiteness of the moderate rational points 273 Observe that it follows from Faltings work on the Mordell conjecture that if the hyperbolic curve under consideration is of genus 2, then the set of rational points, hence also moderate rational points, is finite. The main result of the present paper is as follows [cf. Corollaries 2.6, 2.7]: Theorem A Let k be a number field and (G, o G(k)) an elliptic curve (respectively, a nonsplit torus of dimension one) over k. Write = G \ {o}. [Thus, is a hyperbolic curve over k of type (1, 1) (respectively, (0, 3)).] Then the set of moderate k-rational points of is finite. In order to prove Theorem A, we discuss the center of the image of the pro-l outer Galois action ρ : G k Out( ) associated to. In particular, we prove the following result [cf. Theorem 1.13]: Theorem B Let (g, r) be a pair of nonnegative integers such that 2g 2+ r > 0, l an odd prime number, k a generalized sub-l-adic field [cf. [7, Definition 4.11]], k an algebraic closure of k, and a split [cf. Definition 1.3, (i); Remark 1.3.1, (i)] hyperbolic curve of type (g, r) over k which has no special symmetry [cf. Definition 1.3, (ii); Remark 1.3.1, (ii)]. Write G k = Gal(k/k), for the pro-l geometric fundamental group of [i.e., the maximal pro-l quotient of π 1 ( k k)], ρ : G k Out( ) for the pro-l outer Galois action associated to, Γ = Im(ρ ), and M ) Out( = ( )ab Zl F l. Then the following three conditions are equivalent: (1) It holds that (g, r) {(1, 1), (2, 0)}, and that 1 Aut(M ) is contained in the image of the action G k Aut(M (2) The center Z(Γ ) is isomorphic to Z/2Z. (3) The center Z(Γ ) is nontrivial. 0. Notations and Conventions ) induced by ρ. Numbers. The notation Z will be used to denote the ring of rational integers. If l is a prime number, then we shall write F l = Z/lZ and Z l for the l-adic completion of Z. We shall refer to a finite extension of the field of rational numbers as a number field.
274. Hoshi Profinite Groups. Let G be a profinite group, H G a closed subgroup of G, and G Q a quotient of G. We shall say that H (respectively, Q) is characteristic if every [continuous] automorphism of G preserves H (respectively, Ker(G Q)). We shall write G ab for the abelianization of G [i.e., the quotient of G by the closure of the commutator subgroup of G], N G (H) for the normalizer of H in G, Z G (H) for the centralizer of H in G, and Z(G) = Z G (G) for the center of G. We shall say that G is slim if Z G (J) = {1} for every open subgroup J G of G. Let G be a profinite group. Then we shall write Aut(G) for the group of [continuous] automorphisms of G, Inn(G) Aut(G) for the group of inner automorphisms of G, and Out(G) = Aut(G)/ Inn(G) for the group of outer automorphisms of G. If G is topologically finitely generated, then one verifies easily that the topology of G admits a basis of characteristic open subgroups, which thus induces a profinite topology on the group Aut(G), hence also a profinite topology on the group Out(G). Curves. Let S be a scheme and a scheme over S. Then we shall say that is a smooth curve over S if there exist a scheme cpt which is smooth, proper, geometrically connected, and of relative dimension one over S and a closed subscheme D cpt of cpt which is finite and étale over S such that the complement cpt \ D of D in cpt is isomorphic to over S. Note that, as is well-known, if is a smooth curve over [the spectrum of] a field k, then the pair ( cpt, D) is uniquely determined up to canonical isomorphism over k; we shall refer to cpt as the smooth compactification of and to D as the divisor at infinity of. Let S be a scheme. Then we shall say that a smooth curve over S is a hyperbolic curve [of type (g, r)] over S if there exist a pair ( cpt, D) satisfying the condition in the above inition of the term smooth curve and a pair (g, r) of nonnegative integers such that 2g 2+r > 0, each geometric fiber of cpt S is [a necessarily smooth proper connected curve] of genus g, and the degree of D cpt over S is equal to r. We shall refer to a hyperbolic curve of type (0, 3) as a tripod. Tori. Let S be a scheme and (, o (S)) an abelian group scheme over S. Then we shall write G m,s for the [abelian] group scheme over S which represents the functor over S
Finiteness of the moderate rational points 275 (T S) Γ(T, O T ). We shall say that (, o (S)) is a split torus over S if (, o (S)) is isomorphic to a fiber product of finitely many copies of G m,s over S. We shall say that (, o (S)) is a torus if, étale locally on S, (, o (S)) is a split torus. Thus, one verifies easily that if (, o (S)) is a torus of relative dimension d over S, then the assignment (T S) Hom T -ab.gr. (( S T, o T (T )), G m,t ) where we write Hom T -ab.gr. for the module of homomorphisms of abelian group schemes over T determines an étale local system in free Z-modules of rank d on S. In particular, if, moreover, S is connected, then the above étale local system gives rise to a continuous outer homomorphism π 1 (S) GL d (Z). Note that one verifies immediately that it holds that the torus (, o (S)) is split if and only if this outer homomorphism is trivial. 1. Center of the Outer Galois Image Associated to a Hyperbolic Curve In the present Section 1, we discuss the center [which is necessarily a finite group cf., e.g., [3, Proposition 1.7, (ii)], [3, Lemma 1.8]] of the image of the pro-l outer Galois action associated to a hyperbolic curve over a generalized sub-l-adic field [cf. [7, Definition 4.11]]. In particular, we give, under the assumption that l is odd, a necessary and sufficient condition [cf. Theorem 1.13] for a split [cf. Definition 1.3, (i); Remark 1.3.1, (i)] hyperbolic curve over a generalized sub-l-adic field which has no special symmetry [cf. Definition 1.3, (ii); Remark 1.3.1, (ii)] to have trivial center. In the present Section 1, let (g, r) be a pair of nonnegative integers such that 2g 2 + r > 0, l a prime number, k a field of characteristic zero, k an algebraic closure of k, a hyperbolic curve of type (g, r) over k, and a smooth variety over k [i.e., a scheme which is smooth, of finite type, separated, and geometrically connected over k]. Write G k = Gal(k/k), cpt for the smooth compactification of, fundamental group of, = π 1 ( k k) for the geometric
276. Hoshi ρ : G k Out( ) for the outer Galois action associated to, for the pro-l geometric fundamental group of [i.e., the maximal pro-l quotient of ], and M = ( )ab Zl F l. [Thus, M is equipped with a natural structure of vector space over F l of finite dimension.] If Q is a characteristic quotient of, then we shall write ρ Q : G k Out(Q) for the outer Galois action determined by ρ and Γ [Q] = Im(ρ Q ) Out(Q). Write, moreover, ρ = ρ : G k Out( ) [i.e., the pro-l outer Galois action associated to ] and Γ = Γ [ ] = Im(ρ ) Out( ), Γ Γ (mod-l) = Γ [ ] = Im(ρ ) Out( ), = Γ [M ] Out(M ) = Aut(M ). Theorem 1.1 (Mochizuki) Suppose that k is generalized sub-l-adic [cf. [7, Definition 4.11]]. Then the natural homomorphisms Aut k () Out( ) Out( ) determine isomorphisms of finite groups Aut k () Z Out( )(Γ ) Z Out( )(Γ ). Proof. This follows immediately from [7, Theorem 4.12] [cf. also [8, Corollary 1.5.7]].
Finiteness of the moderate rational points 277 Definition 1.2 Let Q be a characteristic quotient of and α Aut k ( ) an automorphism of over k. Then we shall say that α is Q-Galois-like if the image of α Aut k ( ) via the composite of natural homomorphisms Aut k ( ) Out( ) Out(Q) is contained in Γ [Q] Out(Q). We shall say that α is l-galois-like (respectively, (mod l)-galoislike) if α is -Galois-like (respectively, M -Galois-like). Remark 1.2.1 One verifies immediately from the various initions involved that the natural injection Aut k () Out( ) determines an injection {l-galois-like automorphisms of over k} Z(Γ ). If, moreover, k is generalized sub-l-adic, then it follows from Theorem 1.1 that this injection is, in fact, an isomorphism: Definition 1.3 {l-galois-like automorphisms of over k} Z(Γ ). ( i ) We shall say that the hyperbolic curve is split [cf. [3, Definition 1.5, (i)]] if the divisor at infinity of determines a trivial covering of Spec(k), or, equivalently, the natural action of G k on the set of cusps of is trivial. (ii) We shall say that the hyperbolic curve has no special symmetry [cf. [3, Definition 3.3]] if the following condition is satisfied: Write M g,r for the moduli stack of r-pointed proper smooth curves of genus g over k whose r marked points are equipped with an ordering, (Cg,r cpt M g,r ; s 1,..., s r : M g,r Cg,r cpt ) for the universal family over M g,r, and C g,r = Cg,r cpt \ r i=1 Im(s i). [Thus, C g,r M g,r is a split hyperbolic curve of type (g, r) over M g,r.] Then the specialization homomorphism Aut Mg,r (C g,r ) Aut k ( k k) [obtained by equipping the cusps of k k with some ordering] is an isomorphism. Remark 1.3.1 ( i ) It follows from the inition of a hyperbolic curve that there exists a finite extension K of k such that k K is split. (ii) In the notation of Definition 1.3, (ii), it is well-known [cf., e.g., [2,
278. Hoshi Theorem 1.11]] that the functor over M g,r (S M g,r ) Aut S (C g,r Mg,r S) is represented by a finite unramified [relative] scheme A M g,r over M g,r. Thus, there exists a dense open substack U M g,r of M g,r on which A M g,r is a finite étale covering. Thus, it follows immediately from the inition given in Definition 1.3, (ii), that each hyperbolic curve parametrized by a point of U has no special symmetry. (iii) It follows immediately that the hyperbolic curve has no special symmetry if and only if Aut k ( k k) is isomorphic to the finite group S 3 (if (g, r) = (0, 3)) Z/2Z Z/2Z (if (g, r) = (0, 4)) Z/2Z (if (g, r) {(1, 1), (1, 2), (2, 0)}) {1} (if 2g 2 + r 3). In this situation, if, moreover, is split, then one verifies immediately that Aut k () is isomorphic to the above finite group. Lemma 1.4 Suppose that is split. Then the following hold: ( i ) Let α be an l-galois-like automorphism of over k. Then α induces the identity automorphism on the set of cusps of. (ii) Suppose, moreover, that is of genus zero. Then there is no nontrivial l-galois-like automorphism of over k. Proof. First, we verify assertion (i). Since is split, one verifies immediately that the natural action of Γ, hence also α [cf. our assumption that α is l-galois-like], on the set of conjugacy classes of cuspidal inertia subgroups of is trivial. Thus, assertion (i) follows from the [well-known] injectivity of the natural map from the set of cusps of to the set of conjugacy classes of cuspidal inertia subgroups of. This completes the proof of assertion (i). Assertion (ii) follows immediately from assertion (i), together with the [easily verified] fact that every nontrivial automorphism of over k acts nontrivially on the set of cusps of [cf. our assumption that is of genus
Finiteness of the moderate rational points 279 zero]. This completes the proof of assertion (ii), hence also of Lemma 1.4. Proposition 1.5 Suppose that k is generalized sub-l-adic, and that is split and of genus zero. Then the center Z(Γ ) is trivial. Proof. This follows immediately from Lemma 1.4, (ii), together with Remark 1.2.1. Lemma 1.6 Let O, A be profinite groups; f : O A a homomorphism of profinite groups; G O a closed subgroup of O; α N O (G). Suppose that the following three conditions are satisfied: (1) The kernel of f is pro-l. (2) There exists a positive integer n such that n is prime to l, and, moreover, α n = 1. (3) It holds that f(α) f(g). Then it holds that α G. Proof. Let us first observe that one verifies easily that, to verify Lemma 1.6, we may assume without loss of generality, by replacing O by the closed subgroup of O generated by G and α, that G is normal in O [cf. our assumption that α normalizes G]. Next, let us observe that one verifies immediately from condition (3) that, to verify Lemma 1.6, we may assume without loss of generality, by replacing (O, A) by (f 1 (f(g)), f(g)), that f G, hence also f, is surjective. In particular, the natural inclusion N = Ker(f) O determines an isomorphism N/(N G) O/G, which thus implies that O/G is pro-l [cf. condition (1)]. Thus, it follows immediately from condition (2) that the image of α O in O/G is trivial, i.e., that α G. This completes the proof of Lemma 1.6. Lemma 1.7 Let α Aut k ( ) be an automorphism of over k of finite order. Suppose that α is of order prime to l. Then it holds that α is l-galois-like if and only if α is (mod l)-galois-like. Proof. Let us first observe that since the natural surjection M factors through, the necessity follows from the inition given in Definition 1.2. To verify the sufficiency, suppose that α is (mod l)-galoislike. Write α[ ] Out( ) for the image of α in Out( ). Then
280. Hoshi one verifies immediately that α[ ] centralizes Γ Out( ). Thus, ) Aut(M ) since the kernel of the natural homomorphism Out( is pro-l [cf., e.g., [1, Theorem 6]], by applying Lemma 1.6 in the case where we take (O, A, G, α) in the statement of Lemma 1.6 to be (Out( ), Aut(M ), Γ, α[ ]), we conclude that α[ ] Γ, i.e., that α is l-galois-like. This completes the proof of the sufficiency, hence also of Lemma 1.7. Lemma 1.8 Let (T, o T (k)) be a nonsplit torus of dimension one over k. Suppose that = T \ {o}. [Thus, is a nonsplit tripod over k.] Suppose, moreover, that the following two conditions are satisfied: (1) l is odd. (2) k contains a primitive l-th root of unity. Then the automorphism α Aut k () of over k induced by the automorphism of T given by a a 1 [i.e., by multiplication by 1 if the group operation of T is written additively] is l-galois-like. Proof. Write χ: G k Z for the [necessarily nontrivial] character determined by the nonsplit torus T. Let γ G k be such that χ(γ) is nontrivial. Write I o M for the [uniquely determined] inertia subgroup of M associated to the cusp [corresponding to] o of. Then let us observe that one verifies easily that we have a natural exact sequence of finite G k -modules 1 I o M M T 1. Observe that I o and M T are equipped with natural structures of vector spaces over F l of dimension one. Next, let us observe that it follows immediately from the inition of α that the actions of α on I o, M T are given by multiplication by 1, 1, respectively. On the other hand, it follows immediately from our choice of γ G k, together with condition (2), that the actions of γ on I o, M T are given by multiplication by 1, 1, respectively. Thus, since γ commutes with α, we conclude from condition (1) that the action of α on M coincides with the action of γ on M. In particular, α is (mod l)-galois-like, hence also [cf. condition (1), Lemma 1.7] l-galois-like. This completes the proof of Lemma 1.8.
Finiteness of the moderate rational points 281 Lemma 1.9 Let α Aut k () be an automorphism of over k of order prime to l. Write α cpt Aut k ( cpt ) for the automorphism of cpt determined by α. Then the following hold: ( i ) If α is l-galois-like [i.e., (mod l)-galois-like cf. Lemma 1.7], then α cpt is l-galois-like [i.e., (mod l)-galois-like cf. Lemma 1.7]. (ii) Suppose that one of the following two conditions is satisfied: (1) r 1. (2) α induces the identity automorphism on the set of cusps of, k contains a primitive l-th root of unity, and is split. Then it holds that α is l-galois-like if and only if α cpt is l-galois-like. Proof. Assertion (i) follows immediately from the fact that the natural surjection factors through the natural surjection cpt. Assertion (ii) in the case where condition (1) is satisfied follows immediately in light of Lemma 1.7 from the [easily verified] fact that the natural surjection determines an isomorphism M M cpt cpt. Finally, we verify assertion (ii) in the case where condition (2) is satisfied. Let us first observe that it follows in light of Lemma 1.7 from assertion (i) that it suffices to verify that if condition (2) is satisfied, and α cpt is (mod l)-galois-like, then α is (mod l)-galois-like. Write α[m ] Aut(M ) for the image of α in Aut(M ) and A Aut(M ) for the subgroup of Aut(M ) consisting of automorphisms of M that = Ker(M M ) M cpt., M csp cpt ) of F l-linear induce the identity automorphism on M csp Here, let us observe that [since the module Hom Fl (M homomorphisms M M csp cpt natural homomorphism A Aut(M is of order a power of l] the kernel of the ) is an l-group. Moreover, it follows cpt immediately from condition (2) that α[m ] is contained in A. Next, let us observe that the natural action of Γ (mod-l) on M preserves the subspace M csp M. Moreover, it follows immediately from condition (2) that the resulting action of Γ (mod-l) on M csp is trivial, i.e., that Γ(mod-l) A. Thus, by applying Lemma 1.6 in the case where we take (O, A, G, α) in the statement of Lemma 1.6 to be (A, Aut(M ), Γ (mod-l) cpt, α[m ]), we conclude that α[m ] Γ(mod-l), i.e., that α is (mod l)-galois-like. This
282. Hoshi completes the proof of assertion (ii) in the case where condition (2) is satisfied. Definition 1.10 Let α Aut k () be an automorphism of over k. Then we shall say that α is a hyperelliptic involution if g 1, and, moreover, there exist a proper smooth curve C over k of genus zero and a finite morphism cpt C of degree two over k such that Aut C ( cpt ) is generated by [the automorphism of cpt determined by] α. In particular, a hyperelliptic involution is of order two. Remark 1.10.1 If (g, r) {(1, 1), (2, 0)}, then it is well-known that, in the notation of Definition 1.3, (ii), the image of the unique nontrivial [cf. Remark 1.3.1, (iii)] element of Aut Mg,r (C g,r ) in Aut k () [cf. the fact that is split] is a hyperelliptic involution. Lemma 1.11 Let α Aut k () be a hyperelliptic involution of. Then α acts on ab, hence also M cpt, via multiplication by 1. cpt Proof. Let us first observe that it follows immediately that we may assume without loss of generality, by replacing k by k, that k is algebraically closed. Write J for the Jacobian variety of cpt. Then one verifies immediately from the various initions involved that, to verify Lemma 1.11, it suffices to verify that α acts on J via multiplication by 1. Write j : cpt J for the closed immersion associated to a k-rational closed point of cpt that is preserved by α. [Note that since g 1, there exists a k-rational point of cpt that is preserved by α.] Then since J is generated by the image of j, and [one verifies easily that] the morphism cpt C of Definition 1.10 factors through the composite cpt j j (id J,α J ) J k J J k J J where we write α J for the automorphism of J induced by α, and the third arrow is the group operation of J Lemma 1.11 follows immediately from the [well-known] fact that any morphism from a smooth curve of genus zero to an abelian variety is constant. This completes the proof of Lemma 1.11. Lemma 1.12 Let α Aut k () be a hyperelliptic involution of. Suppose that the following two conditions are satisfied:
Finiteness of the moderate rational points 283 (1) l is odd. (2) 1 Γ (mod-l) Aut(M cpt ). cpt Suppose, moreover, that one of the following two conditions is satisfied: (3) r 1. (4) g = 1, is split, and α induces the identity automorphism on the set of cusps of. Then α is l-galois-like. Proof. First, I claim that the following assertion holds: Claim 1.12.A: If g = 1, then condition (2) implies condition (2) in the case where we take to be k k(ζ l ) where ζ l k is a primitive l-th root of unity. Indeed, Claim 1.12.A follows immediately from the [well-known] fact that 2 M is isomorphic, as a G cpt k -module, to the group of l-th roots of unity of k [cf. our assumption that g = 1], together with the [easily verified] fact that 1 Aut(M ) acts trivially on 2 M cpt. This completes the cpt proof of Claim 1.12.A. Now since α acts on M via multiplication by 1 [cf. Lemma 1.11], cpt Lemma 1.12 in the case where condition (3) (respectively, (4)) is satisfied follows immediately from Lemma 1.9, (ii) (respectively, Lemma 1.9, (ii), together with Claim 1.12.A). This completes the proof of Lemma 1.12. Theorem 1.13 Let (g, r) be a pair of nonnegative integers such that 2g 2+r > 0, l an odd prime number, k a generalized sub-l-adic field [cf. [7, Definition 4.11]], k an algebraic closure of k, and a split [cf. Definition 1.3, (i); Remark 1.3.1, (i)] hyperbolic curve of type (g, r) over k which has no special symmetry [cf. Definition 1.3, (ii); Remark 1.3.1, (ii)]. Write G k = Gal(k/k), for the pro-l geometric fundamental group of [i.e., the maximal pro-l quotient of π 1 ( k k)], ρ : G k Out( ) for the pro-l outer Galois action associated to, Γ = Im(ρ ), and M ) Out( = ( )ab Zl F l. Then the following three conditions are equivalent: (1) It holds that (g, r) {(1, 1), (2, 0)}, and that 1 Aut(M ) is contained in the image of the action G k Aut(M (2) The center Z(Γ ) is isomorphic to Z/2Z. ) induced by ρ.
284. Hoshi (3) The center Z(Γ ) is nontrivial. Proof. First, we verify the implication (1) (2). Suppose that condition (1) is satisfied. Then since [we have assumed that] is split and has no special symmetry, it follows from Remark 1.3.1, (iii), that Aut k () Z/2Z. Write α Aut k () Z Out( ) [cf. Theorem 1.1] for the unique )(Γ nontrivial element of Aut k (). Then it follows from Lemma 1.12, together with Remark 1.10.1, that α Γ, i.e., that α Z(Γ ). In particular, it follows that Z/2Z α Z(Γ ) Z Out( )(Γ ) Aut k () Z/2Z. Thus, condition (2) is satisfied. This completes the proof of the implication (1) (2). The implication (2) (3) is immediate. Finally, we verify the implication (3) (1). Suppose that condition (3) is satisfied. Let us first observe that it follows from Proposition 1.5 that g 1. Next, let us observe that since [we have assumed that] has no special symmetry, if (g, r) {(1, 1), (1, 2), (2, 0)}, then it follows from Remark 1.3.1, (iii), together with Remark 1.2.1, that condition (3) is not satisfied. Thus, we conclude that (g, r) {(1, 1), (1, 2), (2, 0)}. Assume that (g, r) = (1, 2). Then one verifies immediately [cf., e.g., the proof of [3, Proposition 3.2, (i)]] that the unique nontrivial automorphism of over k [cf. Remark 1.3.1, (iii)] acts nontrivially on the set of cusps of. Thus, it follows immediately from Lemma 1.4, (i), together with Remark 1.2.1, that condition (3) is not satisfied. In particular, we conclude that (g, r) {(1, 1), (2, 0)}. Next, let us observe that since (g, r) {(1, 1), (2, 0)}, it follows immediately from Remark 1.3.1, (iii), together with Remark 1.10.1, that Aut k () is generated by a hyperelliptic involution α of, i.e., (Z/2Z ) α = Aut k () Z Out( )(Γ ). Thus, since Z(Γ ) ( Z Out( )(Γ )) is nontrivial [cf. condition (3)], it holds that α Z(Γ ) Γ. In particular, condition (1) follows immediately from Lemma 1.11. This completes the proof of the implication (3) (1), hence also of Theorem 1.13. Proposition 1.14 Let be a finite étale Galois covering over k such that is geometrically connected over k, i.e., that is a
Finiteness of the moderate rational points 285 hyperbolic curve over k. Write Π, Π for the respective geometrically pro-l fundamental groups of, [i.e., the respective quotients of π 1 (), π 1 ( ) by the normal closed subgroups Ker( ) π 1(), Ker( ) π 1( )]. Suppose that induces an outer open injection Π Π. Consider the following three conditions: (1) It holds that Ker(ρ ) Ker(ρ ), or, equivalently [cf. [4, Proposition 25, (i)]], Ker(ρ ) Ker(ρ ). (2) There exists an automorphism α Aut ( ) Aut k ( ) of over, hence also over k, which is nontrivial and l-galois-like. (3) The hyperbolic curve k k Ker(ρ ) over k Ker(ρ ) is not split. Then we have implications (1) (2) = (3). If, moreover,, hence also, is of genus zero, then we have equivalences (1) (2) (3). Proof. First, we verify the equivalence (1) (2). Write Q = Π /Z ( Π ), Q = Π /Z ( Π ). [Thus, Q coincides with the quotient Φ {l} /k ined in [4, Definition 1, (iv)] cf. also [4, Lemma 4, (i)].] Then it follows immediately that one obtains a commutative diagram of profinite groups [cf. also [4, Lemma 4, (i)]] 1 Q Γ 1 1 Q Γ 1 where the horizontal sequences are exact, and the vertical arrows are outer injections whose images are open and normal. Now let us observe that one verifies immediately that condition (2) is equivalent to the following condition: (a) There exist elements γ Q Q, γ \ such that the
286. Hoshi [necessarily nontrivial cf. our assumption that γ action of γ action of γ Q on on ] outer obtained by conjugation coincides with the outer obtained by conjugation. In particular, by multiplying γ Q in (a) with a suitable element of if necessary, it follows that condition (2) is equivalent to the following condition: (b) There exist elements γ Q Z Q ( ). Q, γ \ such that γ Q γ 1 Now since = Q, one verifies immediately that condition (b) is equivalent to the following condition: (c) It holds that Z Q ( ) Q. Moreover, since Z Q ( ) = {1} [cf. [4, Lemma 4, (i)]], condition (c) is equivalent to the following condition: (d) It holds that Z Q ( ) {1}, or, equivalently [cf. [4, Lemma 5]], Z Q ( ) {1}. On the other hand, one verifies immediately that condition (d) is equivalent to condition (1). This completes the proof of the equivalence (1) (2). Next, we verify the implication (3) (1). Suppose that condition (3) is satisfied. Let us first observe that since Ker(ρ ) Ker(ρ ) [cf. [4, Proposition 25, (i)]], we may assume without loss of generality, by replacing k by k Ker(ρ ), that ρ is trivial. On the other hand, by considering the natural action of G k on the set of cusps of, we conclude that condition (3) implies that ρ is nontrivial. In particular, condition (1) is satisfied. This completes the proof of the implication (3) (1). Finally, the implication (2) (3) in the case where is of genus zero follows immediately in light of the inclusion Ker(ρ ) Ker(ρ ) [cf. [4, Proposition 25, (i)]], together with the equivalence (1) (2) from Lemma 1.4, (ii). This completes the proof of Proposition 1.14.
Finiteness of the moderate rational points 287 2. Finiteness of the Moderate Points of Certain Hyperbolic Curves In the present Section 2, we discuss the finiteness of the set of moderate rational points of a hyperbolic curve over a number field. In particular, we prove that the set of moderate rational points of the hyperbolic curve obtained by forming the complement of the origin in an elliptic curve, as well as a nonsplit torus of dimension one, over a number field is finite [cf. Corollaries 2.6, 2.7]. In the present Section 2, we maintain the notation of the preceding Section 1. Suppose, moreover, that k is a number field, which thus implies that k is generalized sub-p-adic for every prime number p. Definition 2.1 Let x be a closed point of. Then we shall say that x is moderate if there exists a prime number p such that x is p-moderate [cf. [5, Definition 2.4, (ii)]]. Proposition 2.2 The set of l-moderate [cf. [5, Definition 2.4, (ii)]] k- rational points of is finite. Proof. This follows immediately in light of the equivalence (1) (3) of [5, Proposition 2.5] from the final portion of [4, Theorem A]. Lemma 2.3 Let x (k) be a k-rational point of and α Aut k () an automorphism of over k. Suppose that x is l-moderate, and that α is l-galois-like. Then x (k) is preserved by α. Proof. Write U for the open subscheme of obtained by forming the complement of [the image of] x in and Out x ( U ) Out( U ) for the group of outer automorphisms of U that preserve the conjugacy class of a cuspidal inertia subgroup associated to x (k). Then one verifies immediately from the various initions involved that the natural open immersion U induces a commutative diagram of profinite groups Γ U Out x ( U ) Γ Out( )
288. Hoshi where the vertical arrows are the natural inclusions. Now since x is l-moderate, it follows immediately from the equivalence (1) (2) of [5, Proposition 2.5], that the upper horizontal arrow Γ U Γ of the above diagram is an isomorphism. On the other hand, since α is l-galois-like, it follows immediately from Remark 1.2.1 that α determines an element of Z(Γ ). Thus, by means of the isomorphism Γ U Γ, we obtain an element of Z(Γ U ). Write β Aut k (U) for the automorphism of U over k that corresponds relative to the isomorphism in the second display of Remark 1.2.1 to this element of ). Then one verifies immediately from the injectivity of the composite Z(Γ U Aut x k(u) Aut k () Out( ) where we write Aut x k(u) Aut k (U) for the group of automorphisms of U over k that preserve the cusp of U corresponding to x that β = α U, which thus implies that α preserves the k-rational point x (k). This completes the proof of Lemma 2.3. Proposition 2.4 In the notation of Proposition 1.14, suppose that k is a number field. Then any of the three conditions (1), (2), and (3) implies the following condition: has no l-moderate k-rational point. Proof. Since [one verifies immediately that] every k-rational point of is not preserved by the α in condition (2), this follows immediately from Lemma 2.3. Theorem 2.5 Let be a hyperbolic curve over a number field k. Suppose that there exists a nontrivial automorphism α Aut k () of over k such that, for all but finitely many prime numbers p, the automorphism α is p-galois-like [cf. Definition 1.2]. Then the set of moderate [cf. Definition 2.1] k-rational points of is finite. Proof. Write S for the set of prime numbers p such that the automorphism α is p-galois-like and Z S (k) for the set of k-rational points of which are p-moderate for some p S. Then let us observe that since [we have assumed that] the complement of S in the set of all prime numbers is finite, it follows immediately from Proposition 2.2 that, to complete the verification of Theorem 2.5, it suffices to verify that Z S is finite. On the
Finiteness of the moderate rational points 289 other hand, it follows immediately from Lemma 2.3 that every element of Z S is preserved by α. Thus, the finiteness of Z S follows immediately from the [well-known] finiteness of the set of fixed points of a nontrivial automorphism of a hyperbolic curve. This completes the proof of Theorem 2.5. Corollary 2.6 Let (T, o T (k)) be a nonsplit torus of dimension one over a number field k. Write = T \ {o}. [Thus, is a nonsplit cf. Definition 1.3, (i) tripod over k.] Then the set of moderate [cf. Definition 2.1] k-rational points of is finite. Proof. Write α Aut k () for the automorphism of over k induced by the automorphism of T given by a a 1 [i.e., by multiplication by 1 if the group operation of T is written additively]. Then it follows from Theorem 2.5 that, to complete the verification of Corollary 2.6, it suffices to verify that α is p-galois-like for all but finitely many prime numbers p. On the other hand, this follows immediately from Lemma 1.8, together with the [easily verified] fact that Gal(k/k(ζ p )) Ker(χ), where we write χ: G k Z for the necessarily nontrivial character determined by the nonsplit torus T and use the notation ζ p k to denote a primitive p-th root of unity, for all but finitely many prime numbers p. This completes the proof of Corollary 2.6. Corollary 2.7 Let (E, o E(k)) be an elliptic curve over a number field k. For a positive integer n, write E[n] E for the subgroup scheme of E obtained by forming the kernel of the endomorphism of E given by multiplication by n. Let D E be a closed subscheme of E such that E[1] (= {o} ) D E[2]. Write = E \ D for the hyperbolic curve over k obtained by forming the complement of D in E. [Thus, is of type (1, 1), (1, 2), (1, 3), or (1, 4).] Then the set of moderate [cf. Definition 2.1] k-rational points of is finite. In particular, the set of moderate rational points of a hyperbolic curve of type (1, 1) over a number field is finite. Proof. Let us first observe that it follows immediately from the equivalence (1) (2 ) of [5, Proposition 2.5], that, to verify Corollary 2.7, we may assume without loss of generality, by replacing k by a suitable finite extension of k, that is split [cf. Remark 1.3.1, (i)]. Write α Aut k () for the automorphism of over k determined by the automorphism of E
290. Hoshi given by multiplication by 1. Now it follows immediately from Theorem 2.5 that, to complete the verification of Corollary 2.7, it suffices to verify that α is p-galois-like for all but finitely many prime numbers p. Next, let us observe that one verifies easily that α is a hyperelliptic involution of and induces the identity automorphism on the set of cusps of. Thus, it follows immediately from Lemma 1.12 [in the case where condition (4) is satisfied] that, to verify the assertion that α is p-galoislike for all but finitely many prime numbers p, it suffices to verify that 1 Aut(M (p) p) E ) (= Aut(E[p](k))) is contained in Γ(mod E Aut(M (p) E ) (= Aut(E[p](k))) for all but finitely many prime numbers p. On the other hand, this follows from [9, Section 4.4, Théorème 3]; [9, Section 4.5, Corollaire to Théorème 5]. This completes the proof of Corollary 2.7. Acknowledgments The author would like to thank Akio Tamagawa for helpful discussions concerning Lemma 1.8 and Corollary 2.6. This research was supported by Grant-in-Aid for Scientific Research (C), No. 24540016, Japan Society for the Promotion of Science. References [ 1 ] Anderson M. P., Exactness properties of profinite completion functors. Topology 13 (1974), 229 239. [ 2 ] Deligne P. and Mumford D., The irreducibility of the space of curves of given genus. Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75 109. [ 3 ] Hoshi., Galois-theoretic characterization of isomorphism classes of monodromically full hyperbolic curves of genus zero. Nagoya Math. J. 203 (2011), 47 100. [ 4 ] Hoshi., On monodromically full points of configuration spaces of hyperbolic curves, The Arithmetic of Fundamental Groups - PIA 2010, 167 207, Contrib. Math. Comput. Sci., 2, Springer, Heidelberg, 2012. [ 5 ] Hoshi., On the kernels of the pro-l outer Galois representations associated to hyperbolic curves over number fields. Osaka J. Math. 52 (2015), 647 677. [ 6 ] Matsumoto M., Difference between Galois representations in automorphism and outer-automorphism groups of a fundamental group. Proc. Amer. Math. Soc. 139 (2011), 1215 1220. [ 7 ] Mochizuki S., Topics surrounding the anabelian geometry of hyperbolic curves, Galois groups and fundamental groups, 119 165, Math. Sci. Res. Inst. Publ., 41, Cambridge Univ. Press, Cambridge, 2003. [ 8 ] Nakamura H., Galois rigidity of pure sphere braid groups and profinite cal-
Finiteness of the moderate rational points 291 culus. J. Math. Sci. Univ. Tokyo 1 (1994), 71 136. [ 9 ] Serre J. P., Propriétés galoisiennes des points d ordre fini des courbes elliptiques. Invent. Math. 15 (1972), 259 331. Research Institute for Mathematical Sciences Kyoto University Kyoto 606-8502, Japan E-mail: yuichiro@kurims.kyoto-u.ac.jp